In an earlier lesson in this unit, students studied the converse of the theorem they study in this lesson. Previously, they proved that a segment which splits two sides of a triangle proportionally will be parallel to the third side. In this lesson, they prove that a line parallel to one side of a triangle splits the other two sides proportionally. They can now conclude that a line is parallel to one side of a triangle if and only if it splits the other two sides proportionally.
In proving the theorem in this lesson, students look for and make use of structure (MP7) as they decide how they will prove that two triangles are similar. Students have the option to use repeated reasoning as they apply the Angle-Angle Triangle Similarity Theorem or they can continue to look for sequences of dilations and rigid motions to prove two triangles are similar.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
- Prove that a segment parallel to one side of a triangle divides the other sides proportionally (in writing).
- Let’s investigate parallel segments in triangles.
- I can explain why a segment parallel to one side of a triangle divides the other sides proportionally.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).